Question

Inverse of f(x)=2 1/2-3 1/3x and what is its x intercept

Answer 1

Find appropriate f(x) first

• 2 1/2=5/2
• 3 1/3=10/3

Now

• y=-10/3x+5/2

So

InterchAnge x and y

• x=-10/3y+5/2

Find y which is the inverse

• x-5/2=-10/3y
• y=-3/10x+5/2×3/10
• y=-3/10x+3/4

x intercept

• 10/3x+3/4=0
• x=-3/4×3/10
• x=-9/40

Graph attached

Answer 2

Answer:

$\textsf{Inverse function}: \quad f^{-1}(x)=\dfrac{3}{4}-\dfrac{3}{10}x$

$x\textsf{-intercept}:\quad\left(\dfrac{5}{2},0\right)$

Step-by-step explanation:

Given:

$f(x)=2\frac{1}{2}-3\frac{1}{3}x$

Rewrite the function so it is a rational function

Convert the mixed numbers to improper fractions:

$\implies f(x)=\dfrac{5}{2}-\dfrac{10x}{3}$

Make the denominators the same:

$\implies f(x)=\dfrac{3 \cdot 5}{3\cdot 2}-\dfrac{2 \cdot 10x}{2 \cdot 3}$

$\implies f(x)=\dfrac{15}{6}-\dfrac{20x}{6}$

Combine:

$\implies f(x)=\dfrac{15-20x}{6}$

The inverse of a function is its reflection in the line y = x

To find the inverse, make x the subject

Replace f(x) with y:

$\implies y=\dfrac{15-20x}{6}$

$\implies 6y=15-20x$

$\implies 6y-15=-20x$

$\implies x=\dfrac{-6y+15}{20}$

$\implies x=\dfrac{15-6y}{20}$

Replace x with $f^{-1}(x)$ and y with x:

$\implies f^{-1}(x)=\dfrac{15-6x}{20}$

If necessary, convert back into the same format as the original function:

$\implies f^{-1}(x)=\dfrac{15}{20}-\dfrac{6x}{20}$

$\implies f^{-1}(x)=\dfrac{3}{4}-\dfrac{3}{10}x$

The x-intercept of the inverse function is the point at which it crosses the x-axis, so when $f^{-1}(x)=0$

$\implies \dfrac{15-6x}{20}=0$

$\implies 15-6x=0$

$\implies 6x=15$

$\implies x=\dfrac{15}{6}=\dfrac{5}{2}$

Therefore, the x-intercept is:

$\left(\dfrac{5}{2},0\right)$